Humanizing
Calculus
Michelle Cirillo
I know how to use the power rule to [find] derivatives, but I don’t know where it came from or who
made it up or why we have to use it—except that
this is what the book and teacher wants us to do.
—Calculus student, spring 2005
T
he student quoted above raises a concern about the way in which some
mathematics topics are still being
presented today. As teachers, we
sometimes present formulas and rules
but do not take the time to talk about the evolution of the mathematics as a human invention.
The history of mathematics can supply the why,
where, and how for many concepts that are studied (Swetz 1995).
Classroom teachers can help their students
develop an appreciation of the invention of calculus in many ways. To address the concerns raised
by the student in the opening quotation, I sprinkle
quotations and anecdotes throughout a calculus
course, some of which I have included in this
article. Stories about the invention of calculus by
Sir Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716) can add a zesty backdrop to your
calculus lessons and thus help students come to see
mathematics as a body of knowledge developed by
human beings.
Vol. 101, No. 1 • August 2007 | Mathematics Teacher 23
Copyright © 2007 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
WHO INVENTED CALCULUS?
On the first day of class, I give this homework
assignment: Find out who invented calculus. A
basic Internet search will turn up the names of
Newton and Leibniz. Students who investigate
further may realize that a long line of mathematicians contributed to the development of calculus.
Beginning with the work of Eudoxus in the fourth
century BCE and spanning about 2000 years, the
groundwork was laid for the invention of calculus.
As Hellman (1998) states, neither Newton nor
Leibniz created calculus “out of the thin air” (p.
42). Rather, because of the contributions made by
other mathematicians, such as Archimedes, Kepler,
Descartes, Fermat, Pascal, and Barrow, the basic
components of their versions of calculus were
already in existence. Any number of mathematicians (if they were alive today) might argue, based
on their accomplishments, that they invented calculus. A possible research project for students would
be to determine the contributions made by the
predecessors of Newton and Leibniz. Students may
even stage a debate among themselves as an activity
or project in the month that follows the Advanced
Placement exam.
CONTEMPORARY GENIUSES
Both Newton and Leibniz are today considered to
be among the greatest geniuses ever to have lived.
Students are usually interested to learn that they
came from vastly different backgrounds. Their
fathers died while Newton and Leibniz were very
young, but the similarities between them end
there. Newton grew up on a farm, and his father
was illiterate, signing his name with an X. Leibniz came from an educated family (Bardi 2006).
His father was a professor of moral philosophy.
Leibniz proudly claimed that he was self-taught,
reading from his father’s library and attending
college at age 14. He earned his bachelor’s and
master’s degrees in only three years and completed
a doctorate in law by the age of 20. Newton, on
the other hand, was less than a shining star as a
youngster, and his mother took him out of school
at age 16 to work on the family farm. He eventually returned to school and attended Cambridge
University. It was during the Great Plague, while
the university was closed, that Newton, working
in isolation, laid the groundwork for his work in
physics, optics, astronomy, and mathematics. Leibniz, on the other hand, whom Frederick the Great
of Prussia dubbed “a whole academy unto himself,” did not begin his serious study of mathematics until he was on a diplomatic mission in Paris
in 1672 (Dunham 1991). It is no surprise that two
contemporary geniuses of this magnitude might
have had an intellectual spat or two.
24 Mathematics Teacher | Vol. 101, No. 1 • August 2007
Newton
Leibniz
THE PRIORITY DISPUTE
The term priority dispute refers to the debate over
who invented calculus. Historians of mathematics
today generally agree that Newton and Leibniz both
invented calculus. Newton is credited as being the
first inventor of calculus (1665–1666), and Leibniz
is recognized as having independently invented
calculus (1675) and as being responsible for its dissemination and for developing the notation that is
most similar to that used in modern calculus texts.
According to Katz (1993), Newton and Leibniz (not
Fermat, Barrow, or their predecessors) are both considered the inventors of calculus for four reasons:
1. They each developed general concepts. Newton
utilized the fluxion (velocity or rate of change)
and fluent (flowing quantity), while Leibniz
used the differential and integral. These ideas
are related to the two basic problems of calculus:
extrema and area.
2. They developed notation and algorithms.
3. They took their respective concepts and applied
the inverse relationship.
4. They used these two concepts and applied them
to previously unsolvable problems.
The most compelling piece of evidence that clears
Leibniz from an accusation of plagiarism actually
came from Newton himself. In his first edition of
Principia (1687), Newton wrote:
In letters which went between me and that most
excellent geometer, G.W. Leibniz, 10 years ago,
when I signified I was in the knowledge of a method
of determining maxima and minima, of drawing
tangents, and the like . . . that most distinguished
man wrote back that he had also fallen on a method
of the same kind, and communicated his method
which hardly differed from mine, except in his form
and words and symbols. (Gjertsen 1986, p. 469)
Clearly, Leibniz did not steal his method from Newton, but by the third edition of Principia (1726), ten
years after Leibniz’s death, Newton had removed
any reference to Leibniz or his work. Perhaps Leibniz would have avoided these accusations had he
d
d
d
d
fg = 
f
g
d
d 
f±
g  ,dx
 dx   dx 
 ( f ± g) =
dx
dx 
 dx
acknowledged the correspondence between himself
and Newton, as Newton had done in the first edition of the Principia.
Although Newton developed his calculus six
years before Leibniz even began his serious study
of mathematics, Newton failed to publish his work.
Newton, however, believed that a scientist’s priority derives from having done the work and not from
the publication of the discovery (Hellman 1998).
This belief produced much conflict and was the
source of consternation for many years to come.
There are various opportunities throughout a
calculus course to bring in Leibniz and Newton. In
the next section, I will provide examples that can be
used when teaching the product and quotient rules
and implicit differentiation.
CALCULUS HISTORY FOR SPECIFIC TOPICS
Derivatives in Early Calculus—The Product and
Quotient Rules
After students learn the basic formulas for differentiation of sums and differences
d
d
d 
f±
g ,
 ( f ± g) =
dx
dx
dx


they should be asked to think about what a product
 d  d 
d
or quotient rule fg
might
f like
g d  2004). It
d =  look
d (Cupillari
dx
±
g ) suggest
= dx d
f ±product
gd , and
 quo  (df dx
is likely that students
will
dx
dx
f dx
±  g ,
  ( f ± g) =
tient rules of the forms:
dx
dx 
 dx
d
 df   d 
 dx
d d f fg
=  d
f
g
=
d
d g  fgd=dx . d f dx dd g 
 dx
dx



(
f
±
g
)
=
f
±
g
  dx gd ,
 dx
 ddx
d 
dx
 dd dx
dd dx
(f ± d
gd)dx
=  f ±
g ,

(
f
±
g
)
=
f
±
g
,
and
f±
gdx
,
 ( f ± g ) = dx
dx 
d
dx
dx
dx
dx f dx 
 dx
d  f  dx d f 
dd
xy
d
=
.d


d f
dxfg g= d=df dx
 dx. gd   d 


dx
dx




g

dd
g
 fgdd=  f
dx ddg f dx
fg
fg ==  dx
f   ggdx   dx 
x
dx
dx
d
x




dx
d ;  dx   dx 
dx
y
d
d xy
Leibniz, himself,
 dx f made dthis assumption
d incorrectly
df xy
d
f
.
d df  the
before correctly
discovering
  =β
dx . and quo. df f  =product
dx
=ff2zg+ bdx
ddy
d
dx
g


tient rules (Cupillari
dated
. In a manuscript
x = 2004).
d
d
dx . g 
 gg; x= dd dx
dx
g
dx 1675,
November 11,
wrote:
g
yd Leibniz
dx
dx dy = y2 z;dx
+ bcgβ 2 .
dx
xy
Let us nowdexamine
whether dx dy is the same
dy
= 2 z whether
+ dbβxy
.
thing as dddv
xy
v
xy, and
dy
= 2; z + bβ . dx/dy is the same
=
d
thing as dψ x ψ
d ;
β2. 2
y dy = 2 zd+xbc
xxdx
;
dd ;; dx dy = 2yz + bcβ .
yy
dydv
= 2=z d+ vbβ;. 2
it may be seen that
yv= z + bz, and x = cz + d;
dv ifdy
d
ψ
ψ =; 2 z + bβ .
then . . . dy
dy==22zzd+ψ
+ bbβ=β..dIn
ψ the same way dx = + cb,
and hence dx dy = 2 z + bcβ 2 . (Child 1920, p. 100)
2
dx dy
2 = 2 z + bcβ .
dx
dxdy
dy==22zz ++ bc
bcββ 2..
With an explanation
of
dv
v the notation (we would
d ;parentheses
use dz in place of b= and
in place of the
dv
v
dv
vψ
=d ;
dvdψ
== dd v ;; dψ
ψ
ddψ
ψ
ψ
ψ
d
f
 d  d 
d


d
f
dx .
fg = is perfectly
f
gaccessible
bar sign), Leibniz’s example
=

dx
 dx   dx 
dx  g 
d
to calculus students. What is surprising about this
g
dx
example is Leibniz’s next statement:
d
f


d
f
dx
But you get the same thing
= if you. work out d xy


dx manner
g
d. . . and it is the
in a straightforward
g
same thing in the case of divisors.
(Child 1920,
dx
x
d ;
p. 100)
y
d xy
Leibniz did not work out the details of this claim.
dy = 2 z + bβ .
Instead, he considered results from the “inverse
x
method of tangents”dand; discovered his error
y
(Cupillari 2004). The mathematics
involved here
is = 2 z + bcβ 2 .
dx dy
not as accessible to beginning calculus students, but
through that investigation,
dy = 2 z Leibniz
+ bβ . realized:
dv
v
=d ;
d
ψ
ψ
Hence it appears that it is incorrect
to say that
dx dy = 2 z + bcβ 2 .
dv dy is the same thing as dvy, or that
dv
v
=d ;
dψ
ψ
although just above I stated that this was the
case, and it appeared to be proved. This is a difficult point. (Child 1920, p. 101)
Rather than going back to his original example,
Leibniz provided a counterexample, using v = x
and y = x. Ten days later, in a manuscript dated
November 21, 1675, Leibniz provided the correct
product and quotient rules. After stating the correct
product rule, Leibniz wrote, “Now this is a really
noteworthy theorem and a general one for
all curves” (Child 1920, p. 107).
Child (1920) points out that, as a logician, Leibniz should have known better than to believe he
proved the product rule by providing a single example. A discussion about what counts as “proof”
could follow, and students could use the correct
product rule to determine that the true value of
d(xy) in Leibniz’s original problem should have
been (3cz2 + 2(bc + d)z + bd)dz. Correcting Leibniz’s
mistake is a fun way to involve students in the
human invention. This example shows students
that “calculus was not created in one sequential,
correct, and ordered way, as it is presented in textbooks” and that “even a mathematician as brilliant
as Leibniz made mistakes when he did not check
his work correctly” (Cupillari 2004, p. 195).
When does (fg)′ = f′g′?
While there are an infinite number of counterexamples to the equation ( fg)′ = f ′g′, there are also an
infinite number of pairs of functions f and g that
satisfy the equation. For the trivial cases where f or
g is the zero function, or if both f and g are constants, then, of course, ( fg)′ = f ′g + fg′ = f ′g′. As
Vol. 101, No. 1 • August 2007 | Mathematics Teacher 25
k2
f ( x ) = Ce k−1
x
k2
x
f ( x ) = Ce k−1
n
 x 
f (x) = K 
n
 x − n ximplicit

Maharam and Shaughnessy (1976) pointed out,
When learning
differentiation, students
f (x) = K 
( fg)′ = f ′g′ is true for any functions f and g, when
can implicitly
a problem that was actu− n 
 xdifferentiate
2
f(x) = C(n – x)–n and g(x) = xn, where C is an arbially posed
of
 xand
 solved by Newton: the problem
fcalculating
( x)= 5
trary nonzero constant and n ≠ 0. One example of
the
tangent
to the cubic curve x3 – ax2 +
2

 x − 2 x 
two such functions is f(x) = 3(2 – x)–2 and g(x) = x2. axyf (–xy)3==5 0. In
a tract known as the Methodus flux x − 2  infinitorum (The Method of
Maharam and Shaughnessy (1976) provide a list
ionum et serierum
xFluxions and Infinite Series), which was written
of other, more interesting pairs of functions for
which the “incorrect product rule” produces corin 1671
but not published until 1736, Newton prex
x
csc x extends this work,
rect answers. Cupillarie (2004)
sented this equation simply as an example, ascrib
y
e x csc
x
showing that an infinite
number
of function pairs
ing no significant meaning to this particular curve.
y
can be found to satisfy the incorrect quotient rule
However,
a curve can be thought of as a path traced

xby a moving point; and, in this case, the curve New f ′ f ′
tonxpresented contains the factors (x – y)(x2 + xy –
=
′
2
 fg fg′′
yax + y ), a line and an ellipse. Surely, Newton’s
=
 g 
interests
in cubic curves of this nature stemmed
g′
y
by letting
from his interest in science. For example, Newton
k2

xo
x
f ( x ) = Cekk2−1
showed that a planet orbits the sun under an
x
 square law of attraction moving, not in a cirxo
f ( x ) = Ce k−1
inverse

yo
cle, but in an ellipse (Hellman 1998). Additionally,
n
 x 
with k ≠ 1 and g(x) =f (ekx
can
hisyo
n generate
work on the tangent line problem stemmed
x.) Students
= K
 xx− n
pairs of functions and
they satisfy both
from
his interest in optics and light refraction

f ( xcheck
) = K that
x
+
xo
x − n  and quotient
the correct and the incorrectproduct
(Larson, Hostetler, and Edwards 1998).

x + xo
rules. The fact that these are recent 2publications
To differentiate x3 – ax2 + axy – y3 = 0, Newton
 x 

y
+
yo
2 is an evolving,
also demonstrates that
substituted x + ẋo for x and y + ẏo for y to get
f ( xmathematics
)= 5
 xx− 2

y + yo
rather than a static, factivity.
( x)= 5

 x − 2
 )3 − a( x + xo
 )2 + a( x + xo
 )( y + yo
 ) − ( y + yo
 )3 = 0.
( x + xo
Differing Approaches
x and Implicit
 )3 − a( x + xo
 )2 + a( x + xo
 )( y + yo
 ) − ( y + yo
 )3 = 0.
( x + xo

Differentiation in xEarly Calculus
The
expansion
3
2
2of2 this3equation
3
2 gives us
2 2





x + 3x xo + 3xx o + x o − ax − 2axxo − ax o
Newton’s fluxional calculus
and Leibniz’s differen2
2
2  2 + axyo
y
− ax 2o2
+ axyo
− 2y−3 −2a3xyxo
axyo
 yo
x+3 +axy
3x+2 xo
+ 3xx
o + x 3o3− ax
tial calculus had different
goals.
Leibniz’s
interests
y
2 2
3 3
2
3
− 3 yy
o −+ yaxyo
o =+0axyo
.  + axyo
  − y − 3 y 2 yo

were more philosophical, and his approach was geo+axy

x
2
2
3
3
metric. Newton’s interest in calculus was primarily
− 3 yy o − y o = 0.
x the more physical aspects of
based on motion and
quantities varying with
Since x3 – ax2 + axy – y3 = 0, then, by substitution,
y time. Newton adopted “a
new approach in which
we are left with
y variables are regarded as
flowing quantities generated by the continuous

 + 3xx 2o2 + x 3o3 − 2axxo
 − ax 2o2
xo
3x 2 xo
motions of points, lines, etc.” (Hollingdale 1989, p.

xo
 + axyo
 + axyo
  2 − 3 y 2 yo

+ axyo
186). Newton called the variable (e.g., x or y) the
2
2
2
3
3
2
2
3
3





yo
fluent and its rate of change or velocity the fluxion,
3x xo
3xxo o− +y xo o =−0.2axxo − ax 2o2
−+
3 yy
 ẏ) to denote the fluxion. Newusing a dot (e.g., ẋ oryo
 + axyo
 + axyo
  2 − 3 y 2 yo

+ axyo
2
2 2
3 3



2 2
ton realized that thexfluxions
themselves can be
Since o is an2 infinitely
we
cast out 
3
x
+
3
xx
o
+
x
2 2 small
3 xo
3 quantity,

2
+ xo
 o+ −axy
y o+ axy
=0.− 3 y y = 0o. − 2axxo − ax o
3x x −− 32yy
axx
2
2
2
3
3
2
2
2
2
considered fluent quantities,
thus having fluxions
these terms, leaving
 + 3xx o+ a+xxyo
 o+ −axyo


3x xo
2 axxo
− ax −
o 3 y yo
x + xo
+ axyo
themselves. These second fluxions of x and y he
23 3
2
2
2
  + −axyo
 

yo
3 yy−+o3axyo

y +Gjersten
yo
3yx 2 x − 2axx++axaxy
+ axy
y−2 yy=o−03.=y0.yo
denoted by ẍ and ÿ. As
(1986) explains
2 2
3 3
,



y + yo to tackle the problems
− 3 yy o − y o = 0.
in The Newton Handbook,
x
3
2
3
 +more
 − 3 y 2 y = 0.
x 2 xmethod
− 2axx will
+ axy
axy





involving the inverse
be
( xrelationship
+ xo ) − a( x between
+ xo ) + afluents
( x + xo )( y +Newton
yo ) − ( yended
+
= 0. but 3the
y yo ) here,
, )if3 =
 )3 − a( x the
 notion
 )( y +familiar
 ) − ( yto
and fluxions, Newton
bit further.
 −Solving
( x +introduced
xo
+ xo
)2 + a( xof+ axo
yo
+yxus
yo
32 x0−2.continue
x2 ax
− 2+axx
+ axy
3 y 2 y = for
0.
3xwe
ay +a axy
=
.
moment:
2
3
2
2 2
3 3
2
2 2

y
 + 3xx o + x o − ax − 2axxo
 − ax xo
3 y − ax ,
x + 3x xo
2
2
2
2
3 xo
2  2 o2
 − 2ax +xay
 ++3axyo

y
x 3 + 3+x axy
xo
xx2o+
+axyo
x 3o3+−axyo
ax
x
−
ax
  2−−2a

y
3
x
This was the “indefinitely small” part by which 2 y3 − 3 y2 yo =
,
3 y 2 − ax .
2
2
3
3





x

+−axy
axyo − y we
− 3get
y yo x
3 yy+small”
oaxyo
− y +periods
oaxyo
= 0.+ of
fluents grew in “indefinitely
2 2
3 3
y 3x 2 − 2ax + ay
− 3 yyby
o the
− ysign
o =o.0The
.
time, and was represented
=
.
2 
y 3x −x 2ax +3ay
y 2 − ax
moment of the fluent x would therefore be ẋo,
=
.
and of the fluent y, ẏo. In this way, it follows that
x
3 y 2 − ax
quantities x and y will become in an indefinitely
small interval x + ẋo and y + ẏo. (Gjertsen 1986,
Note that if we were to use modern methods to difp. 214)
ferentiate implicitly, differentiating the third term
26 Mathematics Teacher | Vol. 101, No. 1 • August 2007
would require use of the product rule. The product
rule is implied with Newton’s method, however.
In contrast, Leibniz was interested in finding an
appropriate notation to represent thoughts and ways
of combining these—a process of computation or
reasoning using symbols. As mentioned earlier, Leibniz developed a product rule and a quotient rule. He
derived other differentiation rules that look similar
to the notation we use today, for example, d(ax) = a
dx (Eves 1983). In addition, Leibniz used the expressions dx and dy to indicate the difference of two
infinitely close values of x and y, respectively, and
dy/dx to indicate the ratio of these two values (Cooke
1997). He also used the symbol ∫ to represent the
idea of sum (Katz 1993) as well as the term function
(Larson, Hostetler, and Edwards 1998). Had Leibniz
used his rules on the problem presented by Newton,
his solution would have looked more like our modern
implicit differentiation. Students can use modern
notation to verify Newton’s solution. Because Leibniz’s notation was considered superior and because
he published his work in 1684, it is his notation,
rather than Newton’s, that is found in today’s modern calculus books.
CONCLUSION
Many examples from history demonstrate that calculus was the creation of human beings. I encourage
you to explore even further. Other topics that can be
visited include the history behind L’Hôpital’s Rule
or the controversy over lack of rigor in Newton’s
and Leibniz’s calculus. This lack of rigor (later filled
in by Cauchy’s delta-epsilon process) was considered
a major problem by eighteenth-century mathematicians. The contributions and rivalry of the Bernoulli
brothers add more fun and humorous stories to the
colorful history of calculus. The two brothers were
contemporaries of Newton and Leibniz and stout
defenders of Leibniz in the priority dispute (Dunham 1991). Multicultural connections can be made
to the Islamic and Indian mathematicians in a discussion about sine and cosine power series.
By including facts and anecdotes about the
mathematicians who created mathematics, teachers can help calculus and mathematics come alive.
Jay Lemke (1990), author of Talking Science, claims
that students are three to four times as likely to
be “highly attentive to ‘humanized’ science talk as
they would be to ‘normal’ science talk in the classroom” (1990, p. 136). Surely the same can be said
for “humanized” calculus.
BIBLIOGRAPHY
Bardi, Jason Socrates. The Calculus Wars. New York:
Thunder’s Mouth Press, 2006.
Child, J. M. The Early Mathematical Manuscripts of
Leibniz. Chicago: Open Court, 1920.
Cooke, Robert. The History of Mathematics: A Brief
Course. New York: John Wiley & Sons, 1997.
Cupillari, Antonella. “Another Look at the Rules of Differentiation.” Primus: Problems, Resources, and Issues in
Mathematics Undergraduate Studies 14 (2004): 193–200.
Dunham, William. Journey through Genius: The Great
Theorems of Mathematics. New York: Penguin
Books, 1991.
———. The Calculus Gallery. Princeton, NJ: Princeton
University Press, 2005.
Eves, Howard. Great Moments in Mathematics (after
1650). Vol. 7. Washington DC: Mathematical Association of America, 1983.
Gjertsen, Derek. The Newton Handbook. New York:
Routledge & Kegan Paul, 1986.
Hellman, Hal. Great Feuds in Science: Ten of the Liveliest
Disputes Ever. New York: John Wiley & Sons, 1998.
Hollingdale, Stuart. Makers of Mathematics. New
York: Penguin Books, 1989.
Katz, Victor J. A History of Mathematics: An Introduction.
New York: Harper Collins College Publishers, 1993.
Larson, Roland E., Robert P. Hostetler, and Bruce H.
Edwards. Calculus with Analytic Geometry. New
York: Houghton Mifflin, 1998.
Lemke, Jay. Talking Science. Westport, CT: Ablex Publishing, 1990.
Maharam, Lewis G., and Edward P. Shaughnessy.
“When Does ( fg)′ = f ′g′?” College Mathematics
Journal 7(1976): 38–39.
Newton, Isaac. The Method of Fluxions and Infinite
Series. 1740. Translated by J. Colson. 2nd ed. Ann
Arbor, MI: Xerox University Microforms, 1976.
Swetz, Frank. “Some Not So Random Thoughts about
the History of Mathematics—Its Teaching, Learning, and Textbooks.” Primus 5 (June 1995): 97–107.
OTHER RESOURCES
courses.science.fau.edu/~rjordan/phy1931/
NEWTON/newton.htm
www-groups.dcs.st-and.ac.uk/~history/
Mathematicians/Newton.html
www-groups.dcs.st-and.ac.uk/~history/
Mathematicians/Leibniz.html ∞
MICHELLE CIRILLO, mcirillo@iastate.
edu, is a former secondary classroom
teacher who is now pursuing her PhD in
mathematics education at Iowa State
University, Ames, IA 50011. Her research interests
include classroom discourse, proof, and the history
of mathematics. Photograph by Lisa Shainker; all rights reserved
Vol. 101, No. 1 • August 2007 | Mathematics Teacher 27